The large charge limit of scalar field theories and the Wilson-Fisher fixed point at $epsilon=0$

We study the sector of large charge operators $phi^n$ ($phi$ being the complexified scalar field) in the $O(2)$ Wilson-Fisher fixed point in $4-epsilon$ dimensions that emerges when the coupling takes the critical value $gsim epsilon$. We show that, in the limit $gto 0$, when the theory naively approaches the gaussian fixed point, the sector of operators with $nto infty $ at fixed $g,n^2equiv lambda$ remains non-trivial. Surprisingly, one can compute the exact 2-point function and thereby the non-trivial anomalous dimension of the operator $phi^n$ by a full resummation of Feynman diagrams. The same result can be reproduced from a saddle point approximation to the path integral, which partly explains the existence of the limit. Finally, we extend these results to the three-dimensional $O(2)$-symmetric theory with $(bar{phi},phi)^3$ potential.